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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dormql (f08cg)

Purpose

nag_lapack_dormql (f08cg) multiplies a general real $m$ by $n$ matrix $C$ by the real orthogonal matrix $Q$ from a $QL$ factorization computed by nag_lapack_dgeqlf (f08ce).

Syntax

[c, info] = f08cg(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)
[c, info] = nag_lapack_dormql(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)

Description

nag_lapack_dormql (f08cg) is intended to be used following a call to nag_lapack_dgeqlf (f08ce), which performs a $QL$ factorization of a real matrix $A$ and represents the orthogonal matrix $Q$ as a product of elementary reflectors.
This function may be used to form one of the matrix products
 $QC , QTC , CQ , CQT ,$
overwriting the result on $C$, which may be any real rectangular $m$ by $n$ matrix.
A common application of this function is in solving linear least squares problems, as described in the F08 Chapter Introduction, and illustrated in Example in nag_lapack_dgeqlf (f08ce).

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

Parameters

Compulsory Input Parameters

1:     $\mathrm{side}$ – string (length ≥ 1)
Indicates how $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{side}}=\text{'L'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the left.
${\mathbf{side}}=\text{'R'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{side}}=\text{'L'}$ or $\text{'R'}$.
2:     $\mathrm{trans}$ – string (length ≥ 1)
Indicates whether $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{trans}}=\text{'N'}$
$Q$ is applied to $C$.
${\mathbf{trans}}=\text{'T'}$
${Q}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension, $\mathit{lda}$, of the array a must satisfy
• if ${\mathbf{side}}=\text{'L'}$, $\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{side}}=\text{'R'}$, $\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dgeqlf (f08ce).
4:     $\mathrm{tau}\left(:\right)$ – double array
The dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$
Further details of the elementary reflectors, as returned by nag_lapack_dgeqlf (f08ce).
5:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double array
The first dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $C$.

Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array c.
$m$, the number of rows of the matrix $C$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array c.
$n$, the number of columns of the matrix $C$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathrm{k}$int64int32nag_int scalar
Default: the second dimension of the arrays a, tau.
$k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraints:
• if ${\mathbf{side}}=\text{'L'}$, ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$;
• if ${\mathbf{side}}=\text{'R'}$, ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.

Output Parameters

1:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double array
The first dimension of the array c will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
c stores $QC$ or ${Q}^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ as specified by side and trans.
2:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: side, 2: trans, 3: m, 4: n, 5: k, 6: a, 7: lda, 8: tau, 9: c, 10: ldc, 11: work, 12: lwork, 13: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

The computed result differs from the exact result by a matrix $E$ such that
 $E2 = O⁡ε C2$
where $\epsilon$ is the machine precision.

The total number of floating-point operations is approximately $2nk\left(2m-k\right)$ if ${\mathbf{side}}=\text{'L'}$ and $2mk\left(2n-k\right)$ if ${\mathbf{side}}=\text{'R'}$.
The complex analogue of this function is nag_lapack_zunmql (f08cu).

Example

See Example in nag_lapack_dgeqlf (f08ce).
```function f08cg_example

fprintf('f08cg example results\n\n');

a = [-0.57, -1.28, -0.39,  0.25;
-1.93,  1.08, -0.31, -2.14;
2.3,   0.24,  0.4,  -0.35;
-1.93,  0.64, -0.66,  0.08;
0.15,  0.3,   0.15, -2.13;
-0.02,  1.03, -1.43,  0.5];
b = [-2.67,  0.41;
-0.55, -3.10;
3.34, -4.01;
-0.77,  2.76;
0.48, -6.17;
4.10,  0.21];
% Compute the QL factorization of a
[a, tau, info] = f08ce(a);

% Compute C = (C1) = (Q^T)*b
%             (C2)
[c, info] = f08cg( ...
'Left', 'Transpose', a, tau, b);

% Compute least-squares solutions by backsubstitution in L*X = C2
[b, info] = f07te( ...
'Lower', 'No Transpose', 'Non-Unit', a(3:6,:), c(3:6,:));

if (info > 0)
fprintf('The lower triangular factor, L, of A is singular,\n');
fprintf('the least squares solution could not be computed.\n');
else
% Print least-squares solutions
[ifail] = x04ca( ...
'General', ' ', b, 'Least-squares solution(s)');
% Compute and print estimates of the square roots of the residual
% sums of squares
rnorm = zeros(2,1);
for j=1:2
rnorm(j) = norm(c(1:2,j));
end
fprintf('\nSquare root(s) of the residual sum(s) of squares\n');
fprintf('\t%11.2e    %11.2e\n', rnorm(1), rnorm(2));
end

```
```f08cg example results

Least-squares solution(s)
1          2
1      1.5339    -1.5753
2      1.8707     0.5559
3     -1.5241     1.3119
4      0.0392     2.9585

Square root(s) of the residual sum(s) of squares
2.22e-02       1.38e-02
```